Pullbacks of Saito-Kurokawa lifts

Abstract

Pullbacks of Siegel Eisenstein series have been studied by Garrett [17], [18], Bocherer [6], [7], Heim [26], and play a key role in the proof of the algebraicity of critical values of certain automorphic L-functions. More generally, one might consider pullbacks of Siegel cusp forms. For example, Ikeda [30] gave a conjectural formula for pullbacks of Ikeda lifts [29] in terms of critical values of L-functions for Spn ×GL2. Also, the Gross-Prasad conjecture [22], [23], [8], [27, §8] would relate pullbacks of Siegel cusp forms of degree 2 to central critical values of L-functions for GSp2 ×GL2 ×GL2. Indeed, Bocherer, Furusawa, and Schulze-Pillot [8] gave an explicit formula for pullbacks of Yoshida lifts [57]. In this paper, we give an explicit formula for pullbacks of Saito-Kurokawa lifts and prove the algebraicity of central critical values of certain L-functions for Sp1 ×GL2. To be precise, let κ be an odd positive integer. Let f ∈ S2κ(SL2(Z)) be a normalized Hecke eigenform and h ∈ S+ κ+1/2(Γ0(4)) a Hecke eigenform associated to f by the Shimura correspondence. Let F ∈ Sκ+1(Sp2(Z)) be the Saito-Kurokawa lift of h. For each normalized Hecke eigenform g ∈ Sκ+1(SL2(Z)), we consider the period integral 〈F|H×H, g× g〉 given by 〈F|H×H, g × g〉 = ∫